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Electron lambdatomography

Edited by Roger D. Kornberg, Stanford University School of Medicine, Stanford, CA, and approved October 14, 2009 (received for review June 12, 2009)
Abstract
Filtered backprojection and weighted backprojection have long been the methods of choice within the electron microscopy community for reconstructing the structure of macromolecular assemblies from electron tomography data. Here, we describe electron lambdatomography, a reconstruction method that enjoys the benefits of the above mentioned methods, namely speed and ease of implementation, but also addresses some of their shortcomings. In particular, compared to these standard methods, electron lambdatomography is less sensitive to artifacts that come from structures outside the region that is being reconstructed, and it can sharpen boundaries.
This article describes a reconstruction method applicable to electron tomography (ET). The rigorous mathematical description of the method and its application to ET is given in ref. 1. Here we concentrate on the functionality of the method in an experimental setting with tests on real ET data. Furthermore, we derive a heuristic explanation for its advantages and guidelines for its usage. Finally, we compare it with the most widely used methods in the field, namely filtered backprojection (FBP) and weighted backprojection (WBP). In this context, it should be mentioned that other reconstruction methods have also been developed and applied to ET. Iterative methods, such as algebraic reconstruction technique (ART) and simultaneous iterative reconstruction technique (SIRT) (2, 3), became practically applicable to ET only after regularization through early stopping. A clever discretization, based on Kaiser–Bessel window functions (blobs), was combined with strongly overrelaxed ART and then applied to ET data in refs. 4–6). Another approach is based on variational regularization where in refs. 7 and 8 relative entropy regularization is applied to ET. For more on these other approaches and their merits, we refer to refs. 9, (section 10.2), 10, and 11.
We begin with a very brief introduction to ET, including a discussion of the various data collection geometries and a mathematical formulation of the structure determination problem in ET. This is followed by a brief outline of the FBP and WBP methods. We then move on to our algorithm, electron lambdatomography (ELT), which is based on twodimensional lambda tomography (12–14). However, ELT is also valid for a broad range of threedimensional data acquisition geometries. It is a method that maintains the main benefits of the FBP and WBP methods, namely speed and ease of implementation, while addressing some of the shortcomings. In particular, ELT is generally less sensitive to artifacts that come from structures outside the region of interest (ROI) than these other methods. We conclude by providing examples of reconstructions obtained by ELT from real and simulated ET data.
Basic Notation.
We now introduce notation used throughout the paper. We let ℝ denote the set of real numbers and ℝ^{+} the set of positive real numbers. The threedimensional space is denoted by ℝ^{3} and the unit sphere in ℝ^{3}, i.e. the set of all orientations in threedimensional space, is denoted by S ^{2}. Furthermore, “:=” in equations will mean “defined as.”
Next, given a function f defined in threedimensional space, the projection P(f) of f is defined as In the mathematics literature, P(f) is called the Xray transform of f. Note that when f is represented by its voxel values in threedimensional space, then P(f)(ω, x) is essentially the sum of the values of f in the voxels that lie on the line through the point x that has direction given by ω.
Finally, in some cases we choose to express formulae explicitly in a specific coordinate system (x,y,z) in ℝ^{3}. In such case we will make use of the following convention: the x axis is parallel the tilt axis and the z axis is parallel to the optical axis of the microscope at 0° tiltangle.
Electron Tomography
Data Collection Geometry.
Many tomographic experimental setups, including ET, yield data recorded on a detector that attains different orientations with respect to the specimen whose internal structure we seek to recover. In the case of ET, each recorded transmission electron microscope (TEM) image is associated with a tilt angle which in turn uniquely specifies an orientation of the specimen with respect to the optical axis of the TEM. Hence, the tilt angle can equally well be reinterpreted as an orientation of the TEM detector with respect to the specimen. The data in a tilt series constitutes a series of TEM images where the tiltangle lies on a curve S of directions in threedimensional space. Below, we explicitly describe this curve for each of the standard data acquisition geometries in ET.
Singleaxis tilting.
Here the specimen is rotated around a single axis perpendicular to the optical axis of the TEM. Then, the curve S is part of a longitude circle on the sphere, i.e. expressed in the (x,y,z) coordinates In Eq. 2, θ_{max} corresponds to the largest tilt angle, which is ≈60°.
Multiaxis tilting.
In this case more than one singleaxis tilt data series are taken. The curve S is given as the union of a number of single axis curves (see Eq. 2) rotated around the z axis. Dual axis tilting corresponds to the case where two singleaxis data sets are taken and fused in the above manner.
Slant tilting.
For fixed 0 < α < π/2, the curve S is the set of angles α radians from the vertical z axis. Hence, S is a latitude circle of the sphere. To get such data, one places the specimen in a plane of angle π/2 − α from the electron beam and rotates the specimen in that plane around a fixed point.
The Model for Image Formation and the Reconstruction Problem.
The idea that data from a TEM image can be interpreted as a “projection of the specimen” (15) underlies all current models for image formation used in ET. This is valid under certain approximations which are discussed in e.g., refs. 1, 9, and 16. Given these approximations, a processed tilt series can be regarded as a finite sample of projections P(f)(ω, x) which are given by the tilt angles ω contained in the curve S and x in the detector plane. Here, the function f: ℝ^{3} → ℝ is the “density” to be recovered and it is proportional to the electrostatic potential, which in turn describes the structure of the molecules in the specimen (9, 17). Furthermore, we can represent the detector plane for tilt angle ω by which is the plane perpendicular to ω through the origin (the “physical” detector plane is a translate of this plane). If we introduce then M_{s} represents the collection of all lines with directions parallel to vectors in S (for each ω ∈ S and x ∈ ω^{⊥}, (ω, x) represents the line parallel ω through x). Therefore, a tilt series will be a finite discrete sampling on M_{s} and the reconstruction problem in ET can be stated as follows.
The reconstruction problem in ET.
After suitable processing of the tilt series, the reconstruction problem in ET can be reformulated as the problem of determining the realvalued function f, or some property thereof (e.g., shape of the structures), from a finite sampling of P(f) on M_{s} in which the directions are contained in the curve S ⊂ S ^{2} are given by the tilt angles.
ET is often compared to medical tomography. However, the ET reconstruction problem has two important difficulties that do not arise in standard reconstruction problems in medical Xray tomography. First, since traditionally only a small portion of the specimen is exposed to the electron beam, we are dealing with a local tomography problem, i.e. the tomographic data originates from a small ROI rather than from the entire specimen. This in turn means that, unless prior knowledge is being used as in ref. 18, structures in the specimen can not be exactly recovered even if one where to have access to noisefree continuum data (19). In ref. 20 this issue is referred to as the unit cell being only partially defined. Second, due to the limited angular range of the tiltangle, all of the standard data acquisition geometries described above yield incomplete data in sense that the curve S yields tilt angles that do not circumscribe the specimen. This is known to introduce severe instability into the reconstruction problem.
To summarize, mathematically exact reconstruction for limited angle region of interest data is not possible even in cases when one has noisefree continuum of data, and the reconstruction problem is severely unstable. However, as we shall see in Theorem 1, certain important features of the specimen can be reconstructed from such data.
Established Analytic Reconstruction Schemes
Both FBP and WBP are wellestablished analytic reconstruction schemes frequently used within the ET community for solving the reconstruction problem in ET.
For tomographic data that satisfies a completeness condition Orlov's condition (ref. 10, section 6.1.1); it can be shown that FBP yields a smooth approximate version of f (see e.g., ref. 21, section 2.34) if the data is complete in a sense that can be made precise. Now, in ET all the data collection geometries mentioned before give rise to data that are not complete (10), hence for such data, FBP will not even recover a smooth approximate version of f.
The WBP approach is the Fourier space formulation of the FBP approach. For the same reasons, WBP will not recover an approximation of f from such ET data; the details are given in ref. 10, section 6.1.1 (see also ref. 21, section 6.2.4).
Practical Considerations.
In the practical usage of FBP and WBP on real ET data, one also has to consider how the reconstruction operator is discretized and how to deal with noisy data.
First, in real applications of the FBP method, it is not desirable to attempt to recover the function exactly. The reason is simple; recovering f from finitely many samples of P(f) on M_{s} is an inherently unstable problem even if the data were complete in the sense of Orlov (10). Hence, a useful reconstruction method must include some kind of regularizing step even in the case of noisefree data, and in the FBP method this is achieved by choosing the filter such that the recovered function is a smooth approximation to f. Thus an accurate and reliable recovery of f using the FBP scheme is possible only for complete data, which for the aforementioned data collection schemes for ET is only possible in special cases in which there is some intrinsic symmetry of the object.
In the particular case of singleaxis tilting, each plane orthogonal to the tilt axis can be dealt with separately. This allows one to reduce the threedimensional reconstruction problem to a stack of twodimensional reconstruction problems. Each of these is, however, a limited angle region of interest problem in the plane due to the limited range of the tilt angle. Still, in the current usages of the FBP method given singleaxis tilting data, one simply chooses the filter in each slice as if one had complete data following the guideline in refs. 19 and 22. In the electron microscopy community, this guideline is frequently referred to as Crowther's criterion (ref. 17, page 316). The reconstruction is also often lowpassfiltered in order to further regularize the solution.
Electron LambdaTomography
ELT is an analytic reconstruction scheme. Therefore, it is based on the same assumptions as the FBP/WBP methods, namely that one can rephrase the reconstruction problem in ET as the problem of recovering a function f from projections P(f) sampled on M_{s}.
In ELT we don't attempt to reconstruct f itself. Instead, we reconstruct a threedimensional structure containing the information about f that can be stably retrieved, which turns out to be certain boundaries of the molecules. Furthermore, the recovered threedimensional structure also shows what is inside and what is outside these boundaries, see Theorem 1 for a more precise statement.
The Reconstruction Operator.
The ELT reconstruction operator that we consider is defined in ref. 1 and is designed for projections sampled on M_{s}. It reads as and we now explain the meaning of the above expression. P_{s} * denotes backprojection operator which is formally defined as where g is a function in data space representing the projected data. Expressed in plain words, P_{s} *(g)( x) is the sum of the data taken over all lines passing through x (since (ω, x − ( x·ω)ω) represents the line parallel ω and passing through x). The backprojection P_{s} * is a fundamental part of the FBP and WBP methods as well as ELT.
The derivative D_{s} ^{2} is a second order differentiation in the detector plane along the tangential direction to the curve S, i.e. where σ is the unit tangent to S at ω ∈ S. D_{s} ^{2} is chosen in this specific direction for mathematical reasons (1, 23).
In contrast to the FBP method, L_{s,μ}(f) is not an approximation of f; L_{s,μ}(f) replaces the filters in FBP/WBP by the simpler filter (−D_{s} ^{2} + μ).
The microlocal regularity principle stated now describes which object boundaries are visible from ET data and what boundaries are wellrecovered by L_{s,μ}(f). For the mathematically precise formulation and proof, see ref. 1.
Theorem 1. One can reconstruct a (molecular) boundary at a point x whenever there is a projection in M_{s} (the continuum data set) along a line (electron path) through x that is tangent to this boundary. Moreover, such “visible” boundaries of f will be boundaries of L_{s,μ}(f). Finally, the recovery of such visible boundaries is mildly illposed so it is possible to stably detect them in practice.
This principle is illustrated our reconstructions in the Examples section and we will discuss this in the last section of the article.
Note that L_{s,μ} is useful for ET because the algorithm reconstructs two important types of features of f. The pure Lambda reconstruction term (derivative term), emphasizes differences in data that occur at boundaries. In other words, the pure Lambda term picks up visible boundaries as given by Theorem 1. However, it does not distinguish interiors from exteriors since the derivative in these areas is typically small. The pure backprojection reconstruction term (μ term), is an averaged version of f. To see this, assume the value of f is large (resp. small) near a point x. Then the data on lines through x will, in general, be large (resp. small) and the μ term P_{s} *(μP)(f) will be large (resp. small). Thus the μ term adds contour to the reconstruction and allows one to distinguish objects from their surrounding. One can see this mathematically by noting that μP_{s} *P(f)( x) is a convolution of f with c/ x on a cone representing the lines in the limited data set (see refs. 1 and 23 for specific calculations). The sum in Eq. 4 defining L_{s,μ}(f) will therefore sharpen boundaries and highlight interiors of objects, as we will show using simulated and real data.
Practical Considerations.
So far, we have introduced two important parameters: μ and the width of the derivative kernel. In the actual implementation of L_{s,μ} in the ET setting, the D_{s} ^{2} operator in Eq. 4 is evaluated using a filter that is a smoothed version of the second derivative (a smoothed central second difference), and the halfwidth of the filter is determined by the noise characteristics of the data, the sensitivity of the detectors and the contrast of the specimen as given in the discussion on “Reconstruction Protocol” below.
The other important parameter is μ. In ref. 14, a paradigm to choose μ is given where a feature is selected and μ is chosen so that the L_{s,μ} reconstruction is closest to flat inside a specific feature. This is possible, in general, because the pure Lambda reconstruction term, Eq. 7 curves down inside regions and the pure backprojection term Eq. 8, term curves up. The first author's student T. Bakhos has tested this, and it works well with low noise data and as long as there is only one main region of interest in the reconstruction region. However, in ET, noise and sparsity of data suggest that more ad hoc methods are better.
Because the ET data are so noisy, we convolve on the detector plane in the direction perpendicular to σ (see Eq. 6), and this brings up a third parameter, the kernel width of this convolution, which for singleaxis tilting is just averaging over slices. As with the derivative kernel, this width is correlated with the noise characteristics of the data. This corresponds to the normal way to regularize FBP or WBP reconstruction. A starting point for setting the width of these kernels could be calculated according to Crowther's Criterion on each slice. This step can be done either slice by slice or after the fact, in 3D, for the whole reconstruction. The reconstruction method is fast enough to allow extensive experiments to calculate optimal parameters.
Advantages of ELT
The FBP/WBP methods have been extensively used over the years and they are well known to deliver a decent result when reasonable parameters are given. The improvement of ELT over FBP/WBP is nevertheless clear and based on the intrinsic differences of the methods. Mathematically, FBP and WBP require complete data to accurately reconstruct, but ET data are not complete. ELT being local, does not require complete data (although it will be difficult for any algorithm to image the invisible singularities described in Theorem 1).
One can see the advantages of locality in several ways. If one compares the convolution kernels for ELT and FBP as done in figure 1 of ref. 1. The ELT kernel is local—it is zero away from the origin, but the FBP kernel is not. In fact, the FBP kernel in that figure has oscillations on the interval where the ELT kernel is zero that are ≈7% of the maximum amplitude. This illustrates the fact that ELT needs only data through the ROI to reconstruct the structures but FBP needs all data on the detector plane (which is not given in ET). Comparing point spread function (PSFs) in Fig. 1, one sees the ELT reconstruction operator has a much more localized PSF than FBP. The “X” or wings at the end of the angular range on both the FBP PSF and the ELT PSF are expected in any limited angle backprojection algorithm. In addition, because the PSF is less localized in the FBP case, the signal spread out and away from the real signal into the surrounding, causing a dilution of the actual signal relative to the background. Thus there is a higher chance to lose a weak signal altogether in the FBP case.
Examples
In this section we compare reconstructions obtained by FBP and ELT. We start out with simulated twodimensional data to illustrate the properties of the ELT operator. Next, we move on to real ET data.
Simulated TwoDimensional Data.
This experiment illustrates the fundamental properties of ELT. First, we compare ELT and FBP reconstructions of a single disk (the top two pictures in Fig. 2). Then, we compare ELT and FBP reconstructions for the same disk plus two objects outside the ROI (the bottom two pictures in Fig. 2). This will show the effects of locality of the ELT operator. The ROI is a disk of radius two units and the phantom in the top pictures in Fig. 2 consists of one disk of radius 0.5 units centered at the origin (see Fig. 2 for the data geometry). Parameters are chosen to be essentially the same as the ones we used in our experiments with real data.
The FBP reconstruction in the top left picture of Fig. 2 shows very dark triangular troughs to the left and to the right of the disk that could mask details near the disk. Although the triangles are visible in the ELT reconstruction in the bottom right picture in Fig. 2, they are less pronounced than in the FBP reconstruction.
In the bottom two reconstructions in Fig. 2 we see the effects of shadowing from far away as well as nearby densities. The reconstructions in those bottom pictures include two disks outside the ROI of radius 0.5 centered at (2.6,0) and (−2.6,0) and density 25 times that of the origin centered disk. These disks affect both the ELT and FBP reconstructions since they affect data through the ROI. At the left and right boundaries of both reconstructions are triangular troughs from these added disks, but the triangles are much darker in the FBP reconstruction than the ELT reconstruction. To only have small troughs is important because deep troughs might mask smaller nearby objects. Furthermore, the lambda boundaries of the center disk are better defined. Taken together this leads to less disturbances in the structures themselves in the ELT density.
Real ET data.
We tested the behavior of ELT versus FBP on a real in vitro biological sample of tobacco mosaic virus (TMV) supplied by J. Butler (MRC laboratory for molecular biology, Cambridge, UK). The rationale for choosing this sample was that the structure of the TMV is wellknown. The FBP algorithm used on this data is the optimized version used at the Department of Cell and Molecular Biology at the Karolinska Institute.
The TMV is made up of a helical cylinder with an outer diameter of ≈18 nm and an inner cylindrical hole along the helical axis with a diameter of ≈4 nm (24, 25). The size and regularity of the TMV deems it a suitable sample for a simple comparison– even at the rather low resolution inherent in this particular experiment at ≈7.5 nm resolution. The contrast between the protein (and RNA) mass and the uniform cylindrical nature of the hole are enough to enable the visualization of the hole despite it being smaller than the resolution of 7.5 nm.
Specimen preparation.
The concentration of TMV particles was 3 mg/mL and the colloidal gold markers added for alignment had a diameter of ≈10 nm. The gold markers were coated with BSA (Amersham AuroProbe EM protein A G10) and washed to remove unbound BSA. The specimen was placed on carboncoated Quantifoil R2/2 grids that were glow discharged on both sides. A Vitrobot was used for the vitrification process at 100% humidity with a 2 s blotting time. After vitrification, we obtained a specimen consisting of TMV particles embedded in a slab of vitrified aqueous buffer with a thickness slightly less than 115 nm.
TEM imaging protocol.
The imaging was done at Sidec AB with a FEI Tecnai Polara with a FEG at 300 kV. The detector was a Gatan UltraScan 1000 with a CCD of 14 μm pixel size giving 2,048 × 2,048 pixels per image and a sensitivity of about five counts per electron. The magnification was calibrated to 19,830× giving a pixel size after magnification of 0.5757 nm. The lowdose tilt series was collected at 10 μm underfocus following the singleaxis tilting scheme with 65 lowdose images collected at every second degree. The total dose used for the lowdose tilt series was 1230 e^{−}/nm^{2}, which gives an average dose of 18.92 e^{−}/nm^{2} (or 6.27 e^{−} per pixel) in one image in the lowdose tilt series. After the lowdose tilt series had been recorded, a postimage was collected at considerable higher dose at 0° tiltangle, but still at 10 μm underfocus, enabling a better view of the imaged area of the specimen.
TEM images of the TMV sample are shown in Fig. 3. The 200 × 200 pixel square region insert in Fig. 3 is overcasting an area slightly to the left in the postimage. This square region shows the projection of the ROI that is reconstructed. It contains two vertical and five horizontal TMV particles crossing the ROI and that we expect to see in the ensuing 3D reconstructions. The reason for choosing the ROI in such an area is to see how the ELT and FBP reconstructions are compromised in a crowded region. In addition we have an enlarged version of the ROI insert at the top right and lower right. The lower right shows the same area from the lowdose nottilted specimen with 200 e^{−}/nm^{2}. The whole Fig. 3 image spans 2,048 × 2,048 pixels with 0.5757 nm pixel size, thus covering 1,179 nm in each direction.
Data preprocessing.
The images in the lowdose tilt series were aligned using the colloidal gold markers. The average alignment error over all the tilts was <0.75 nm (i.e. about 1.3 pixels). From the aligned images we then extracted a data set which in turn served as input data for the 3D reconstructions of the ROI. The ROI has a size of 200 × 200 × 200 voxels and it was calculated from an image support of 300 × 300 pixels from the aligned images in the lowdose tilt series.
Reconstruction protocol.
Crowther's criterion applied to this data dictates using a reconstruction kernel with a halfwidth of 4 nm in FBP. In order to compensate for the loss in resolution due to missing data, following ref. 26, the kernel halfwidth is increased by ≈50%. Furthermore, the low signaltonoiseratio in data motivates further increase in the kernel halfwidth to 7.5 nm.
For ELT, the halfwidths of the convolution kernel representing D_{s} ^{2} in Eq. 4 (the derivative kernel) and the convolution representing the averaging over slices (slice averaging kernel) are related to the signaltonoiseratio in the data. This in turn is related to the sensitivity of the detector, the contrast in the specimen, and the dose. In general one can use larger halfwidths to smooth the reconstruction more when the signaltonoise ratio in the data decreases. To prescribe a specific protocol for choosing the halfwidth would require extensive testing on a large number of specimens. However, it turns out that the halfwidths can be chosen close to the above value of 7.5 nm used for the FBP. We can however account for the anisotropy in resolution caused by limited data and we settle for a halfwidth of 6 nm for the derivative kernel and 7 nm for the slice averaging kernel.
The TEM data of TMV is represented in Figs. 3. Note that it is not easy to see the centrally placed hole even in the highdose postimage shown in Fig. 3. This is due partly to induced beam damages and partly due to the large defocusinduced extra contrast shadowing the fainter low density hole. The hole is however visible in the 3D reconstructions.
Fig. 4 shows a planar crosssection of the FBP and ELT reconstructions of the TMV. In this figure, the emphasis is put on the two TMV particles that can be seen passing vertically through the ROI insert in Fig. 3. The two virions pass on top of the five others, and in Fig. 4 we show a 1.15nmthick slice through the tomograms showing the central part of the virions. The diameter of the TMV particle can be estimated to be just below 18 nm for both virions (the side of the square image is 200 pixel, or 115 nm), and the hole between 4.5 and 5 nm. These values agree with the published structural parameters for TMV (24, 25) and are used as benchmarks to justify our choices of halfwidths described in the reconstruction protocol above. Although the cylindrical hole is visible in both FBP (left) and the ELT reconstructions, the boundaries are more prominent in ELT. Furthermore, it is clear in Fig. 4 that the FBP density is more cluttered than the ELT one, and that e.g. the virion is more connected in the ELT density.
In Fig. 5 we show a 200 × 200 × 160 voxel subregion of the ROI reconstructed by FBP and ELT. The subregion is viewed along the beam direction having the tiltaxis horizontal. The thresholds used for contouring the two tomograms have each been selected so that the structural parameters (outer diameter and diameter of the inner cylindrical hole) related to the virions in the tomograms agree with the published ones. Still, it's clear that the ELT tomogram is easier to interpret due to the obviously lower clutterlevel. The virions are also more disconnected in the FBP tomogram when this threshold is used for contouring. If we decrease the threshold level, the background clutter makes the analysis difficult. One can also see the increased clutter in FBP versus ELT when one rotates the 3D reconstructions even though the FBP reconstructions have been optimized. If the FBP reconstructions are smoothed more, boundaries fade; ELT can smooth more and still show boundaries since boundaries are emphasized in ELT.
Application of Theorem 1 and Conclusions
The principle in Theorem 1 is that boundaries tangent to lines in the data set are easier to reconstruct than boundaries not tangent to any line in the data set. This is illustrated in each of our reconstructions above. In the simulated reconstructions, the angular range of tilts is ±60° from the z axis. Therefore, according to this principle, the visible boundaries will be those on the left and right sides of the circles, and these are exactly the boundaries that are most clearly visible in both reconstructions in Fig. 2. Neither FBP nor ELT show the other, invisible, boundaries particularly well.
One can see from each of the reconstructions how the ELT reconstructions highlights the visible boundaries compared with the FBP reconstructions. For example, the thin cylinder wall of the TMV is more emphasized in the ELT reconstruction in Fig. 4 than in the FBP.
In summary, our reconstructions show that ELT emphasizes visible boundaries, even with noisy data. This allows one to smooth more to get rid of clutter. ELT is local so objects outside the ROI affect reconstructions less than in FBP. Finally, ELT is straightforward to adapt to many data acquisition geometries (1, 23).
Acknowledgments
We thank Sara Sandin and LarsGöran Öfverstedt for acquiring the TMV tiltseries data and Wenner Gren Stiftelserna and Sidec, Stockholm, for support that has greatly aided this research. E.T.Q. was supported by National Science Foundation Grants 0456858 and 0908015 and a Tufts University Faculty Research Award.
Footnotes
 ^{1}To whom correspondence may be addressed. Email: todd.quinto{at}tufts.edu

Author contributions: E.T.Q., U.S., and O.O. designed research; E.T.Q., U.S., and O.O. performed research; E.T.Q. and U.S. contributed new reagents/analytic tools; E.T.Q., U.S., and O.O. analyzed data; and E.T.Q. and O.O. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.
References
 ↵
 ↵
 Schoenmakers R,
 Perquin RA,
 Fliervoet TF,
 Voorhout W
 ↵
 ↵
 ↵
 Marabini R
 ↵
 Frank J
 Carazo JM,
 Herman GT,
 Sorzano COS,
 Marabini R
 ↵
 ↵
 ↵
 ↵
 Öktem O
 ↵
 Norln L,
 Skoglund U,
 Öktem O
 ↵
 Vainberg EI,
 Kazak IA,
 Kurozaev VP
 ↵
 ↵
 ↵
 Erickson HP,
 Klug A
 ↵
 Frank J
 Hawkes PW
 ↵
 Frank J
 Penczek PA,
 Frank J
 ↵
 ↵
 Nattererg F
 ↵
 Hawkes PW
 Hoppe W,
 Hegerl R
 ↵
 Natterer F,
 Wbbeling F
 ↵
 Uhlmann G
 Faridani A
 ↵
 Quinto ET,
 Bakhos T,
 Chung S
 ↵
 Davies DB,
 Saenger W,
 Danyluk SS
 Holmes KC
 ↵
 Butler PJG
 ↵
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